In their 1992 paper, Lin and co-workers [26] discuss various aspects of Bessel beams and their generation. They present a plot, reproduced below in figure 4.7, that compares the transverse intensity profile of a real Bessel beam with a theoretical plot.
Figure 4.7: Transverse intensity profile of a Bessel beam (dark) against a theoreticalJ2 0 pro-
file, published by Linet al. in 1992 [26]. Note that the intensity minima of the experimental beam do not all go to zero, with the offset larger near to the core.
In the plot it can be seen that around the beam core, the inter-ring intensity minima of the experimental beam are generally greater than zero. Moreover, the minimum values appear to increase towards the core of the beam. In the paper, Lin assumed these elevated minima could be explained by the poor modular transfer function of the photographic film they used to record the beam’s cross-sectional intensity profile. While this is a plausible explanation, in fact these non-zero minima are a common feature of experimentally produced Bessel beams. The transverse intensity profiles presented in the paper by McQueen et al. [6], for example, demonstrate similarly displaced intensity minima. Tatarkova et al.
[27] reported being able to generate a Bessel beam whose intensity minima deviated in a characteristic manner which we now refer to as tilt. A transverse intensity profile from Tatarkova’s paper is reproduced in figure 4.8.
In the context of Bessel beams, the term tilt originates from Tatarkova’s observation that a suitably perturbed Bessel beam might offer a form oftilted washboard potential for small particles. The tilted washboard potential is a concept that occurs across the sciences,
4.5. Tilted Bessel beams 102
Figure 4.8: Transverse intensity profile from the paper by Tatarkovaet al. [27]. The dis- placement of the intensity minima increases in almost even steps towards the core, creating a radial washboard potential for small particles.
and it was our hope that atilted Bessel beam could offer a possible system for convenient study of the phenomenon. A washboard potential, shown in figure 4.9, can be thought of as a sequence of potential energy wells, superimposed on a larger energy gradient.
Net particle flow
Energy
Figure 4.9: Illustration of thetilted washboard potential. Particles, due to their Brownian motion, canhopover the barriers between adjacent wells. If the potential landscape istilted
the barriers are lower on one side of each well and the direction of overall particle motion is biased. Here particles are biased to move towards the left.
Each well is characterised by having a lowered barrier on one side. For Brownian parti- cles, if the wells are suitably shallow this can lead to a bias in the overall particle motion. This is related to the concept of Brownian ratchets [28]. Optical Brownian ratchets have
been observed and demonstrated by Lee et al. [29, 30, 31]. Normally, Brownian ratchets imply a temporal modulation of the potential energy landscape. In the case of a tilted Bessel beam, a bias in the movement of particles can be observed without the need for temporal modulation.
In her paper, Tatarkova claims that a tilted Bessel beam can be generated by illuminating the axicon with a slightly converging Gaussian beam. She then claims that the degree of tilt can be altered by slowly varying the degree of the convergence. The paper includes no further discussion of the tilt, or how to generate it. In the next section, I verify Tatarkova’s claims with a simple experiment.
4.5.1
Investigating tilted Bessel beams
To investigate the phenomenon of tilted Bessel beams further, I constructed a simple ex- periment, as illustrated in figure 4.10. All optical elements were mounted on translation stages, with the lens before the axicon,L2, and the CCD camera additionally mounted on incremented rails. By varying two parameters: the position of the lensL2 and the position of the camera, I was able to build up a library of Bessel beam profiles. In this case the focal length ofL1,fL1was 300mm and the focal length ofL2,fL2was 25mm. The default
inter-lens spacing wasfL1+fL2= 325mm. At this default position lensL2 was 5cm from
the back of the axicon. For the data presented here, the CCD camera was held at a fixed point, 5cm from the camera. The position of lens L2 was moved in 1cm increments. The effect on the Bessel beam formed at the CCD camera can be seen in figure 4.11.
It is clear that at the default setting, ∆L2 = 0, tilt is already being generated. This suggests that tilt may be at least in part caused by spherical aberrations. A computational model is being developed to explore this issue in more detail. From the sequence of images it is clear that the tilt varies as a function of the displacement of the second lensL2 along the beam axis.
In a larger optical train, it is possible that the generation of tilt may be enhanced by the presence of further aberrations. It is common to re-size Bessel beams after an axicon using a system of lenses, as shown in figure 4.12. Misalignment of these lenses along the beam propagation axis may also induce tilt. I note from Martirosyanet al. [32] that the
4.5. Tilted Bessel beams 104 AXICON COLLIMATION OPTICS Dx STEERING MIRRORS CAMERA Dx’ IR LASER L2 L1 COMPUTER
Figure 4.10: Experimental set-up for producing and studying tilted Bessel beams. The 25mm lens prior to the axicon was mounted on a sliding rail. Moving it along the direction of propagation was found to perturb the Bessel beam, inducing tilt.
far-field optical distribution produced by an axicon is not in fact a well-formed ring but has a distinctive fringed structure, which they ascertain is partially due to “distortion of the diffracting beam at the axicon tip due to the laser beam diffraction limit”. Such effects may also affect the formation of total intensity nulls in a re-sized Bessel beam.
4.5.2
Particle dynamics in a tilted Bessel beam
My initial charge was to replicate Tatarkova’s work, with the goal of observing biased particle movement due to a tilted Bessel beam. Early results were positive and I was able to create a strongly tilted Bessel beam. Small 2.3µm silica spheres were allowed to settle to the bottom of a sample cell. They were then illuminated from below with the tilted Bessel beam as shown in figure 4.13.
The spheres were observed to percolate down the effective radial washboard potential at a constant rate into the beam core. Because the core is much brighter, the radiation pressure was sufficient to elevate the beads to the top of the sample, where they were observed to gather in a cloud. Because of the reconstructive properties of the core of the Bessel beam [18, 19] the movement of a particle into the core of the beam at the bottom of the sample does not affect particles that have already been elevated.
Figure 4.11: Experimental demonstration of the controlled generation oftilt. As the second collimation lens in the collimation unit is moved along the direction of beam propagation, the degree of tilt increases until the Bessel beam is effectively destroyed. Displacements of lensL2 from the optimal collimation position are ∆L2 =(a)-3cm, (b)-2cm, (c)-1cm, (d)0cm (e) +1cm (f)+2cm (g)+3cm.
4.5.3
Particle dynamics in an untilted Bessel beam
During these studies I noticed that the 2.3µm silica microspheres did not necessarily need a tilted potential in order for them to migrate towards the core. Initially this did not make a great deal of sense, since one might expect that an applied tilt in the potential (thereby lowering the energy barrier on one side of a given ring) would be required to observe a bias in the motion of the particles in a particular direction. At this point my focus shifted and I concentrated on trying to elucidate why particles in anuntilted beam still migrated towards the beam core. I suspected that the preferential movement towards the beam core was due
4.6. Computer modelling 106